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Properties

Label 2-1386-77.54-c1-0-5
Degree $2$
Conductor $1386$
Sign $-0.862 - 0.506i$
Analytic cond. $11.0672$
Root an. cond. $3.32675$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−3.72 − 2.15i)5-s + (1.43 − 2.22i)7-s + 0.999i·8-s + (−2.15 − 3.72i)10-s + (−1.15 + 3.10i)11-s − 1.00·13-s + (2.35 − 1.21i)14-s + (−0.5 + 0.866i)16-s + (−1.66 − 2.88i)17-s + (−1.61 + 2.79i)19-s − 4.30i·20-s + (−2.55 + 2.11i)22-s + (−2.86 + 4.96i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.66 − 0.961i)5-s + (0.541 − 0.840i)7-s + 0.353i·8-s + (−0.680 − 1.17i)10-s + (−0.348 + 0.937i)11-s − 0.277·13-s + (0.628 − 0.323i)14-s + (−0.125 + 0.216i)16-s + (−0.404 − 0.700i)17-s + (−0.370 + 0.641i)19-s − 0.961i·20-s + (−0.544 + 0.450i)22-s + (−0.597 + 1.03i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.862 - 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1386\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 11\)
Sign: $-0.862 - 0.506i$
Analytic conductor: \(11.0672\)
Root analytic conductor: \(3.32675\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1386} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1386,\ (\ :1/2),\ -0.862 - 0.506i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4978347930\)
\(L(\frac12)\) \(\approx\) \(0.4978347930\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (-1.43 + 2.22i)T \)
11 \( 1 + (1.15 - 3.10i)T \)
good5 \( 1 + (3.72 + 2.15i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.00T + 13T^{2} \)
17 \( 1 + (1.66 + 2.88i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.86 - 4.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.74iT - 29T^{2} \)
31 \( 1 + (2.77 - 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 2.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.45T + 41T^{2} \)
43 \( 1 - 4.14iT - 43T^{2} \)
47 \( 1 + (2.77 + 1.60i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.65 + 9.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.98 - 1.72i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.42 - 12.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.165 - 0.286i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.84T + 71T^{2} \)
73 \( 1 + (-7.44 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.9 + 8.04i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + (-2.41 - 1.39i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.890120067973474041396865912219, −8.821217337946741358905397419741, −8.055683776844017432195206486656, −7.41688409769796541965836784651, −7.03247379311364197119719428429, −5.42625058136816454961843030273, −4.68166276342095189305715076453, −4.19110912920725659454683569741, −3.30836263493209937397558870013, −1.54234750247730956398741145235, 0.16223298753946067526997130692, 2.31280437611287242231844672372, 3.06882204155402133020437572292, 4.06560667067115827732478397331, 4.72784085361237518795327295321, 6.00616779132401404879490822767, 6.61048528905344800442470439797, 7.84413672752186676528615227458, 8.144905382842185191102833320456, 9.158576404988485483240779251464

Graph of the $Z$-function along the critical line